Three Dimensional Motion A rigid body free to move within a reference frame will, in the general case, have a simultaneous combination of rotation and translation. In three-dimensional space, there may be rotation about any axis and translation that can be resolved into components along three axes.
Plane Motion In a plane, or two- dimensional space, rigid body motion becomes a combination of simultaneous rotation about one axis (perpendicular to the plane) and also translation resolved into components along two axes in the plane. Planar motion of a body occurs when all the particles of a rigid body move along paths which are equidistant from a fixed plane
Translation All points on the body describe parallel (curvilinear or rectilinear) paths. A reference line drawn on a body in translation changes its linear position but does not change its angular orientation. Rectilinear Translation Curvilinear Translation
Fixed Axis Rotation The body rotates about one axis that has no motion with respect to the “stationary” frame of reference. All other points on the body describe arcs about that axis. A reference line drawn on the body through the axis changes only its angular orientation. When a rigid body rotates about a fixed axis, all the particles of the body, except those which lie on the axis of rotation, move along circular paths
General Plane Motion When a body is subjected to general plane motion, it undergoes a combination of translation and rotation, The translation occurs within a reference plane, and the rotation occurs about an axis perpendicular to the reference plane.
Definition of the DOF The number of degrees of freedom (DOF) that a system possesses is equal to the number of independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time. Note that DOF is defined with respect to a selected frame of reference. xAxA yAyA xBxB YBYB θBθB
DOF of a Rigid Body in a 2D Plane If we constrain the pencil to always remain in the plane of the paper, three parameters are required to completely define its position on the paper, two linear coordinates (x, y) to define the position of any one point on the pencil and one angular coordinate (θ) to define the angle of the pencil with respect to the axes. The minimum number of measurements needed to define its position is shown in the figure as x, y, and θ. This system o has three DOF.
DOF of a Rigid Body in a 2D Plane Note that the particular parameters chosen to define the position of the pencil are not unique. A number of alternate set of three parameters could be used. There is an infinity of sets of parameters possible, but in this case there must be three parameters per set, such as two lengths and an angle, to define the system’s position because a rigid body in plane motion always has three DOF.
DOF of a Rigid Body in 3D Space If the pencil is allowed to move in a three-dimensional space, six parameters will be needed to define its position. A possible set of parameters that could be used is three coordinates of a selected point, (x, y, z), plus three angles (θ, φ, ρ). Any rigid body in a three- dimensional space has six degrees of freedom. Note that a rigid body is defined as a body that is incapable of deformation. The distance between any two points on a rigid body does not change as the body moves. ϕ θ ρ
Links A link is a rigid body that possesses at least two nodes for attachment to other links. Binary link - one with two nodes. Ternary link - one with three nodes. Quaternary link - one with four nodes.
Joints A joint is a connection between two or more links (at their nodes), which allows some motion, or potential motion, between the connected links. Joints (also called kinematic pairs) can be classified in the following ways: 1.By the type of contact between the elements, line, point, or surface. 2.By the number of degrees of freedom allowed at the joint. 3.By the type of physical closure of the joint: either force or form closed. 4.By the number of links joined (order of the joint).
Joint Classification by Type of Contact The links joint by a joint may have a surface contact (as with a pin surrounded by a hole), a line contact (as with two cams), or a point contact (as with a ball on a flat surface). the term lower pair describes joints with surface contact. and the term higher pair to describe joints with point or line contact. The main practical advantage of lower pairs over higher pairs is their better ability to trap lubricant between their enveloping surfaces. This is especially true for the rotating pin joint. A pin joint therefore is preferred for low wear and long life, even over its lower pair cousin, the prismatic or slider joint.
The Six Lower Pair Joints Revolute (R) joint Prismatic (P) joint Helical (H) joint Cylindrical (C) joint Spherical (S) joint Flat (F) joint
Joint Classification by Type of Contact: Surface Contact (Lower Pairs) The pin joint or revolute (R) joint and the translating slider or prismatic (P) joints are the only lower pairs usable in a planar mechanism. Revolute (R) joint Prismatic (P) joint Helical (H) joint Cylindrical (C) joint Spherical (S) joint Flat (F) joint
Joint Classification by Type of Contact: Surface Contact (Lower Pairs) The screw or helical (H) joint, the cylindrical (C) joint, the spherical (S) joint, and flat (F) joint are also lower pair joints used in spatial (3-D) mechanisms. These joint pairs may be obtained from a combination of the R and P pairs. Revolute (R) joint Prismatic (P) joint Helical (H) joint Cylindrical (C) joint Spherical (S) joint Flat (F) joint
Joint Classification by Type of Contact: Surface Contact (Lower Pairs) In an (H) joint, motion of either the nut or the screw with respect to the other results in helical motion. If the helix angle is made zero, the nut rotates without advancing and it becomes the revolute (R) joint. If the helix angle is made 90 degrees, the nut will translate along the axis of the screw, and it becomes the prismatic (P) joint. Revolute (R) joint Prismatic (P) joint Helical (H) joint Cylindrical (C) joint Spherical (S) joint Flat (F) joint
Joint Classification by Number of Allowed DOF A more useful means to classify joints is by the number of degrees of freedom that they allow between the joint links. This is equal to the number of independent parameters that need to be specified to completely describe the location of the one of the links if the other link is held fixed Revolute (R) joint Prismatic (P) joint Helical (H) joint Cylindrical (C) joint Spherical (S) joint Flat (F) joint
Joint Classification by Number of Allowed DOF: One DOF Joints The pin joint or revolute (R) and the translating (prismatic) slider joint (P) are 1 DOF joints because they allow only one degree of freedom between the joint links. These are also referred to as full joints (i.e., full = 1 DOF) and are lower pairs. The (R) and (P) joints are both contained within (and each is a limiting case of) the helical (H) joint. The helical joint is achieved by a screw and nut arrangement. Revolute (R) joint 1 DOF Prismatic (P) joint 1 DOF Helical (H) joint 1 DOF
Joint Classification by Number of Allowed DOF: Two DOF Joints 2 DOF Joints allow two simultaneous independent, relative motions, between the joined links. These joint are sometimes referred to as a “half joint.” Example of these joints are the cylindric (C) lower pair joint, and the pin in slot and the cam roll-slide higher pair joints. Cylindrical (C) joint 2 DOF Roll – Slide Cam joint 2 DOF Roll – Slide Pin in Slot joint 2 DOF
Joint Classification by Number of Allowed DOF: Two DOF Joints Note that if you do not allow the two links in a roll-slide joint to slide, perhaps by providing a high friction coefficient between them, you can “lock out” the translating (Δx) freedom and make it behave as a full joint. This is then called a pure rolling joint and has rotational freedom (Δθ) only. A common example of this type of joint is the automobile tire rolling against the road. In normal use there is pure rolling and no sliding at this joint. Friction determines the actual number of freedoms at this kind of joint. It can be pure roll, pure slide, or roll-slide. Roll – Slide Cam joint 2 DOF
Joint Classification by Number of Allowed DOF: Three DOF Joints The flat (F) and the spherical, or ball- and-socket joint are examples of a three-freedom joints. These two pairs are lower pairs because they have surface contact The flat joint allows two translational and one angular independent motions. The spherical joint allows three independent angular motions between the two links joined. This joystick or ball joint is typically used in a three-dimensional mechanism, one example being the ball joints in an automotive suspension system. Spherical (S) joint 3 DOF Flat (F) joint 3 DOF
Joint Classification by the type of physical closure A form-closed joint is kept together or closed by its geometry. A pin in a hole or a slider in a two- sided slot are form closed. In contrast, a force- closed joint, such as a pin in a half-bearing or a slider on a surface, requires some external force to keep it together or closed. This force could be supplied by gravity, a spring, or any external means. There can be substantial differences in the behavior of a mechanism due to the choice of force or form closure. The choice should be carefully considered. In linkages, form closure is usually preferred, and it is easy to accomplish. But for cam-follower systems, force closure is often preferred.
Joint Classification by the number of links joined (order of the joint) The simplest joint combination is when two links are joint. This produces a joint order of one. Joint order is defined as the number of links joined minus one. As additional links are placed on the same joint, the joint order is increased on a one-for-one basis. Joint order has significance in the proper determination of overall degree of freedom for the assembly.
Kinematic Chains, Mechanisms and Machines A kinematic chain is defined as an assemblage of links and joints, interconnected in a way to provide a controlled output motion in response to a supplied input motion. A mechanism is defined as a kinematic chain in which at least one link has been “grounded,” or attached, to the frame of reference (which itself may be in motion). A machine is a collection of mechanisms arranged to transmit forces and do work. Closed kinematic chain
Open and Closed Mechanisms and Kinematic Chains Kinematic chains or mechanisms may be either open or closed. A closed mechanism will have no open attachment points or nodes and may have one or more degrees of freedom. An open mechanism of more than one link will always have one or more degree of freedom, and requires as many actuators (motors) as it has DOF. A common example of an open mechanism is an industrial robot. An open kinematic chain of two binary links and one joint is called a dyad. Open kinematic chain Closed kinematic chain Open Mechanism Closed Mechanism Dyads
DETERMINING DEGREE OF FREEDOM OR MOBILITY OF MECHANISMS
Degrees of Freedom of Planar Mechanisms A.Mobility of one planar link = 3 B.Mobility of L planar links = 3L C.Mobility of (B) when joint by J 1 one DOF joints = 3L – 2J 1 D.Mobility of (C) when joint by J 2 two DOF joints = 3L – 2J 1 – J 2 E.Mobility of (D) with one grounded link = 3(L – 1) – 2J 1 – J 2 M = 3(L −1) − 2J 1 − J 2 where: M = degree of freedom or mobility L = number of links J 1 = number of 1 DOF (full) joints J 2 = number of 2 DOF (half) joints Kutzbach’s Mobility Criterion for Planar Mechanisms
Degrees of Freedom of Spatial Mechanisms M = 6(L −1) − 5J 1 − 4J 2 − 3J 3 − 2J 4 − J 5 where: M = degree of freedom or mobility L = number of links J 1 = number of 1 DOF joints J 2 = number of 2 DOF joints J 3 = number of 3 DOF joints J 4 = number of 4 DOF joints J 5 = number of 5 DOF joints Kutzbach’s Mobility Criterion for Spatial Mechanisms
Mechanisms and Structures The degree of freedom of an assembly of links completely predicts its character. There are only three possibilities. a)M > 0 → mechanism, links will have relative motion. b)M = 0, → structure, no relative between links is possible. c)M < 0, → preloaded structure, no relative between links is possible and some stresses may be present. L = 4 J 1 = 4 J 2 = 0 M = 3(4-1) – 2×4 – 0 = 1 L = 3 J 1 = 3 J 2 = 0 M = 3(3-1) – 2×3 – 0 = 0 L = 2 J 1 = 2 J 2 = 0 M = 3(2-1) – 2×2 – 0 = -1
Assignment Chapter 2 Problems: 2-8, 2-15 and 2-21
Paradoxes Because the Kutzbach’s criterion pays no attention to link sizes or shapes, it can give misleading results in the face of unique geometric configurations. The arrangement shown is known as an “E-quintet,” and it has (DOF = 0) according to Kutzbach’s criterion. Under certain link length conditions, the constant distance constraint imposed by one of the links becomes redundant, and the E-quintet becomes capable of 1 DOF motion. L = 5 J 1 = 6 J 2 = 0 M = 3(5-1) – 2×6 – 0 = 0
Paradoxes If no slip occurs in the cam mechanism shown, Kutzbach’s equation predicts zero DOF. If the two cams take the shape of cylindrical disks, this linkage does move (actual DOF = 1), because the center distance, or length of link the ground link, is exactly equal to the sum of the radii of the two wheels at any time during the motion. The ground link constant length constraint becomes redundant if the two cams take the shape of cylindrical disks, and pinned on their respective centers. L = 3 J 1 = 3 J 2 = 0 M = 3(3-1) – 2×3 – 0 = 0
Inversion A mechanism was defined as a kinematic chain with one of its links grounded. An inversion of a mechanism is obtained by releasing the grounded link and grounding a different link from the original kinematic chain. The number of possible inversions of a mechanism is equal to its number of links, and all inversions have the same mobility or DOF.
The Four Bar Linkage The four bar linkage is one of the simplest mechanism for single-degree-of-freedom controlled motion. It appears in various disguises such as the slider-crank and the cam-follower. It is in fact the most commonly used device in machinery. It is also extremely versatile in terms of the types of motion that it can generate. Simplicity is one mark of good design. The fewest parts that can do the job will usually give the least expensive and most reliable solution. Thus the four bar linkage should be among the first solutions to motion control problems to be investigated
The Four Bar Mechanism A four bar mechanism is obtained by grounding one of the links in the four bar linkage. Four different mechanism inversions may be obtained from the same four bar linkage, all with DOF = 1 L = 4 J 1 = 4 J 2 = 0 M = 3(4-1) – 2×4 – 0 = 1
Inversions of the Four Bar Mechanism Crank Rocker 1 (GCRR) Crank Rocker 2 (GCRR) Double Crank (GCCC) Drag Link Double Rocker (GRCR)
The Grashof Condition on 4 Bar Linkage’s Rotatability The Grashof condition is a simple relationship that predicts the rotation behavior or rotatability of a fourbar linkage based only on the link lengths. Let : L = length of longest link S = length of shortest link P = length of one remaining link Q = length of other remaining link In order for the crank to be pass through point A without locking, the sum of the lengths of the crank link and the ground link (L+S) must be shorter that the sum of the lengths of the two other links (P + Q). L P Q S A S + L ≤ P + Q
The Grashof Condition on 4 Bar Linkage’s Rotatability Let : L = length of shortest link S = length of longest link P = length of one remaining link Q = length of other remaining link Then if : S + L ≤ P + Q the linkage is Grashof class I linkage and at least one link will be capable of making a full revolution with respect to the ground plane. S L P Q S + L ≤ P + Q
The Grashof Condition on 4 Bar Linkage’s Rotatability Based on the relationship between (S + L ) and (P + Q), the following Grashof classes exist: (S + L < P + Q): Grashof Class I linkage. At least one of the links is capable of making full rotation relative to the other links (S + L > P + Q): Grashof Class 2 linkage. None of the links is capable of making full rotation relative to the other links. (S + L = P + Q): Grashof Class 3 linkage. At least one of the links is capable of making full rotation relative to the other links. will have “chang points” twice per revolution of the input crank when the links all become colinear. At these change points the output behavior will become indeterminate.
Motions of Grashof Class I Four Bar Linkage Inversions The motions obtained from the four inversions of of a Grashof Class I four bar linkage, are as follows Ground either link adjacent to the shortest and you get a crank-rocker. Ground the shortest link and you get a double-crank Ground the link opposite the shortest and you will get a Grashof double-rocker, in which both links pivoted to ground oscillate and only the coupler makes a full revolution. S + L < P + Q
Motions of Grashof Class II Four Bar Linkage Inversions None of the links can fully rotate relative to an adjacent link. All inversions will be triple- rockers S + L > P + Q Triple Rocker #1 (RRR1) Triple Rocker #2 (RRR2) Triple Rocker #3 (RRR3) Triple Rocker #4 (RRR4)
Motions of Grashof Class III Four Bar Linkage Inversions All inversions will be either double-cranks or crank- rockers but will have “change points” twice per revolution of the input crank when the links all become colinear. At these change points the output behavior will become indeterminate. At these colinear positions, the linkage behavior is unpredictable as it may assume either of two configurations. S + L = P + Q
Transformations of Four Bar Linkage The basic four bar linkage is a loop of four links joint by four revolute joints. If we relax the constraint that restricted us to only revolute joints, we can transform this basic linkages to a wider variety of mechanisms with greater usefulness. There are several transformation rules that we can apply to planar kinematic chains as discussed next
Transformations of Four Bar Linkage Rule 1: Revolute Joints -> Prismatic Joints Revolute joints can be replaced by prismatic joints with no change in DOF, provided that at least two revolute joints remain in the loop.
Transformations of Four Bar Linkage Rule 2: Full to Half Joints with Link Removal Any full joint can be replaced by a half joint, but this will increase the DOF by one. Removal of a link will reduce the DOF by one. The combination of the two rules above will keep the original DOF unchanged.
Cam Follower Intermittent Motion Mechanisms Intermittent motion is a sequence of motions and dwells. A dwell is a period in which the output link remains stationary while the input link continues to move. There are many applications in machinery that require intermittent motion. The cam-follower variation on the four bar linkage is often used in these situations.
Geneva Mechanisms This is also a transformed four bar linkage in which the coupler has been replaced by a half joint. The input crank (link 2) is typically motor driven at a constant speed. The Geneva wheel is fitted with at least three equispaced, radial slots. The crank has a pin that enters a radial slot and causes the Geneva wheel to turn through a portion of a revolution. When the pin leaves that slot, the Geneva wheel remains stationary until the pin enters the next slot. The result is intermittent rotation of the Geneva wheel.
Ratchet and Pawl The arm pivots about the center of the toothed ratchet wheel and is moved back and forth to index the wheel. The driving pawl rotates the ratchet wheel (or ratchet) in the counterclockwise direction and does no work on the return (clockwise) trip. The locking pawl prevents the ratchet from reversing direction while the driving pawl returns. Both pawls are usually spring- loaded against the ratchet. This mechanism is widely used in devices such as “ratchet” wrenches, winches, etc.
Linear Geneva Mechanism This mechanism is analogous to an open Scotch yoke device with multiple yokes and has linear translational output. It can be used as an intermittent conveyor drive with the slots arranged along the conveyor chain or belt. It also can be used with a reversing motor to get linear, reversing oscillation of a single slotted output slider.